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Inverse Spectral Problem With Low Regularity Refractive Index

Kewen Bu, Youjun Deng, Yan Jiang, Kai Zhang

Abstract

This article investigates the unique determination of a radial refractive index n from spectral data. First, we demonstrate that for piecewise twice continuously differentiable functions, n is not uniquely determined by the special transmission eigenvalues associated with radially symmetric eigenfunctions. Subsequently we prove that if n \in M is twice continuously differentiable functions(or continuously differentiable functions with Lipschitz continuous derivative), then n is uniquely determined on [0,1] by all special transmission eigenvalues when supplemented by partial a priori information on the refractive index.

Inverse Spectral Problem With Low Regularity Refractive Index

Abstract

This article investigates the unique determination of a radial refractive index n from spectral data. First, we demonstrate that for piecewise twice continuously differentiable functions, n is not uniquely determined by the special transmission eigenvalues associated with radially symmetric eigenfunctions. Subsequently we prove that if n \in M is twice continuously differentiable functions(or continuously differentiable functions with Lipschitz continuous derivative), then n is uniquely determined on [0,1] by all special transmission eigenvalues when supplemented by partial a priori information on the refractive index.
Paper Structure (9 sections, 10 theorems, 150 equations, 3 figures)

This paper contains 9 sections, 10 theorems, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical setup and summary of major result
  3. A counterexample of uniqueness and preliminaries
  4. A counterexample of uniqueness
  5. Liouville transform
  6. Key functions
  7. Proof of Theorem \ref{['thm:uniqueness']}
  8. Proof of theorem \ref{['thm:uniqueness_C1']}
  9. Conclusion and Future Work

Key Result

Theorem 2.1

Assume that $n\in C_p^2[0,1]$ with $L$ intervals and $n\in \mathcal{M}$. Assume further that on each interval, it satisfies $0 < n_* < n(r) < n^*$, $|n'(r)|<\tilde{n}^{*}$, $|n"(r)|<\tilde{n}^{**}$. Suppose the values $\{\hat{\delta}_l\}_{l=1}^{L}$ and the value of $n$ on $[\alpha, 1]$ are known, an then the special transmission eigenvalues uniquely determine $n$.

Figures (3)

  • Figure 1: Diagram of the refractive index in Theorem 2.1.
  • Figure 2: The figure of closed curve $\gamma_1$.
  • Figure 3: The figure of $S_2$ bounded by $\gamma_1$ and $\gamma_2$.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.1: HW80
  • Lemma 4.1
  • ...and 13 more